We report NuSTAR observations of a sample of six X-ray weak broad absorption line (BAL) quasars. These targets, at z = 0.148-1.223, are among the optically brightest and most luminous BAL quasars known at z <1.3. However, their rest-frame ≈2 keV luminosities are 14 to >330 times weaker than expected for typical quasars. Our results from a pilot NuSTAR study of two low-redshift BAL quasars, a Chandra stacking analysis of a sample of high-redshift BAL quasars, and a NuSTAR spectral analysis of the local BAL quasar Mrk 231 have already suggested the existence of intrinsically X-ray weak BAL quasars, i.e., quasars not emitting X-rays at the level expected from their optical/UV emission. The aim of the current program is to extend the search for such extraordinary objects. Three of the six new targets are weakly detected by NuSTAR with ≲ 45 counts in the 3-24 keV band, and the other three are not detected. The hard X-ray (8-24 keV) weakness observed by NuSTAR requires Compton-thick absorption if these objects have nominal underlying X-ray emission. However, a soft stacked effective photon index (Γeff ≈ 1.8) for this sample disfavors Compton-thick absorption in general. The uniform hard X-ray weakness observed by NuSTAR for this and the pilot samples selected with